## Abstract

Nozzle facilities, which can generate high Mach number flows, are the core portions of the supersonic wind tunnel. Different from traditional fixed nozzles, a flexible nozzle can deform to designed contours and supply steady core flows in several Mach numbers. Due to the high-quality demands from the thermo-aerodynamic testing, the deformation of the flexible nozzle plate should be carefully designed. This problem is usually converted into the large deformation problem of a cantilever with movable hinge boundary conditions. In this paper, a generalized variational method is established to analyze the deformation behavior of the flexible nozzle. By introducing axial deformation constraint and Lagrange multiplier, an analytical model is derived to predict the deformed morphology of the flexible plate. Finite element analyses (FEA) of a single-jack flexible nozzle model is performed to examine the predicted deformations and reaction forces. Furthermore, the large deformation experiments of an elastic cantilever with a movable hinge connection are carried out to simulate the scenarios in supersonic flexible nozzle facility. Both the FEA and experimental results show high accuracy of current theoretical model in deformation predictions. This method can also serve as a general approach in the design of flexible mechanisms with movable boundaries.

## 1 Introduction

Wind tunnels [1] are the facilities to supply aeronautical environments for thermo-aerodynamic testing. As a kernel portion in wind tunnels, the nozzle component [2] transforms the flow from reservoir condition to testing condition with a designed Mach number. In traditional fixed nozzle facilities, the contours of the nozzle plates are fixed, which means it can only generate a stream with a fixed Mach number. Therefore, the nozzles should be changed frequently according to the demands of various Mach number flows from aircraft testing. The fixed-flexible and full flexible nozzles [3,4] are developed to break these limits, where the flow Mach number is adjustable in one single nozzle. There are two common methods to achieve this purpose, i.e., flexible-wall [5,6] and asymmetric-sliding block nozzle [7,8] designs. The former adjusts the flow Mach numbers via the opposite nozzle plates symmetrically deforming to the designed profile curve and the latter only manipulates the plate at one side. The simplest flexible nozzle is the single-jack nozzle [4,6,9] as shown in Fig. 1(a). By geometry symmetry, this system can be simplified as an elastic cantilever hinged to a retractable rigid rod (jack), whose opposite end is hinged to a fixed point. During operations, the elongation or shortening of the jack leads to plate deformation which leads to jack rotation synchronously. At the design stage, it is important to establish the relationship between the rod elongation and the deformed plate contour. This mechanical model can be treated as a large deformation beam with a movable hinge boundary.

At present, the common approach in designing flexible nozzle is finite element analyses (FEA) [10–12], which cost a large amount of time for geometry and motion modeling. In the aspect of theories, most of the existing models are derived from empirical equations and engineering simplifications, which are not suitable for the scenarios of large deformation [5,13,14]. For large deformation beam problems, the optional approaches include small parameters perturbation method [15,16], elliptic integral method [17–19], and shooting method [20–23]. Actually, all these methods involve solving the deformation process during jack motions. In this paper, a generalized variational principle is established, which can calculate the beam deflection and boundary variation without complex iterations. In Sec. 2 the basic equations and their extensions for cantilever beam hinged to skew rod are derived. Then, the theoretical model is verified by FEA and experiments in Secs. 3 and 4, respectively. Finally, the work is concluded in the last section.

## 2 Analytical Model

*x*,

*y*), as shown in Fig. 1(b). Here, we introduce an assumption that there is no axial deformation for the beam structure, which means its length remains unchanged during deformation. Hence, we can establish the potential energy functional with length restriction as

*w*denotes the deflection function of the beam, $x\xaf$ is the horizontal coordinate of the hinge point after deformation, the superscript “ʹ” is a derivative symbol standing for “d/d

*x*”, and

*λ*denotes the Lagrange multiplier. Based on the principle of minimization potential energy, the first-order variation of Π equals to zero, which leads to

*w*with respect to

*x*can be expressed by its first and second orders as

*w*can also be expressed in the similar way. Hereby, the general solution to Eq. (4) can be expressed in the form of the Taylor series as

*x*

_{0}at the original point, the fourth-order derivative of

*w*can be obtained by Eq. (4),

*S*

_{0}and Δ

*S*denote the original length and elongation of the jack, respectively and

*θ*is the rotation angle of the jack. Simultaneous Eqs. (3) and (8)–(10), the displacement function, the coordinates of the right endpoint, and the rotation angle of the jack are obtained.

*θ*

_{0}is the initial inclination angle of the jack.

## 3 Finite Element Analyses

### 3.1 The Cantilever Model With One Perpendicular Jack.

In this section, the commercial software abaqus is employed to simulate the models in Fig. 1. A four-node shell element with reduced integration is used to model the cantilever beam. The translator, a type of connector element in abaqus, is selected to model the jack, which can realize the translation and rotation movements of the rigid body.

*φ*to denote the rotation angle at the right end of the beam

*M*

_{0}can also be expressed as

The mechanical and geometric parameters used in the finite element model are listed in Table 1. *E* and *ν* represent Young’s modulus and Poisson's ratio, respectively. Two representative geometry configurations (*α* = 0.5 and 0.75) and three tensile loads (*ɛ*_{app} = 30%, 40%, and 50%) are modeled, respectively. All the comparisons of deformations predicted by analytical analyses (ANA) and FEA are demonstrated in Figs. 2(a) and 2(b). Similarly, the deformations corresponding to compressive loads of −30%, −40%, and −50% are shown in Figs. 2(c) and 2(d). The results demonstrate that the current analytical model can predict the deformed configurations with high accuracy. And it has the ability to describe the boundary movements as well as the jack rotation at once.

In addition to the overall deformation, several important parameters are also compared in the case of *α* = 0.75. Figures 3(a)–3(c) show the reaction force (*P*), the rotation angle (*θ*) of the jack, and the tangent angle (*φ*) at the right end of the beam in several tensile loads. It shows that the forces predicted by the analytical model are smaller than the results from FEA. Generally, the stiffness matrix derived in FEA is larger than that from elastic theory, which leads to a larger reaction force. Similarly, for compressive loads, the corresponding *P*, *θ,* and *φ* are also compared in Figs. 3(d)–3(f), receptivity. The minus sign before *ɛ*_{app} denotes the shortening (compressive) of the jack. It is apparently that the rotation angle of the jack is nonlinear with the applied loading, and its change rate accelerates with the growth of jack elongation (or shortening). But in contrast, the tangent angle at the right end of the beam varies linearly with the jack elongation (or shortening).

### 3.2 The Model With One Slant Jack.

The cantilever hinged to a slant jack is also constructed by FEA to further verify the analytical model. The initial inclination angle *θ*_{0} and the length ratio *α* are set as 30 deg and 0.75, respectively. Other parameters remain the same as those in Sec. 3.1. The beam deflections with applied loads of 30%, 40%, and 50% are depicted in Fig. 4(a), respectively. The straight lines with triangle and circle signs represent the results from ANA and FEA, respectively. The results from FEA and theoretical models show good agreements with each other. With the increasing of applied loads, the force predicted by the analytic model and further deviates from that of FEA, as shown in Fig. 4(b). Different from the situation in perpendicular jack, the accumulation errors in rotation angle (Fig. 4(c)) dominate the calculation errors in reaction force, which beat the influence of stiffness matrix deviation. Due to the initial slope of the jack, it first rotates clockwise and then changed to counterclockwise with the increasing of applied load as shown in Fig. 4(c). Similarly, the tangent angle of the beam end increases approximately linear with the applied load as shown in Fig. 4(d). A similar case with the initial inclination angle of −30 deg is also modeled. The predictions of beam deflection, jack reaction force, jack rotation angle, and the tangent angle of the beam end are compared with those from ANA in Fig 5, respectively. It is notable that, the jack rotation angle is approximately linear with the increasing of the applied load, which is different from the previous two models. The selection of the jack initial angle not only affects the rotation direction, but also dominates the change rate of the angle-load curve.

## 4 Experiments

Several simple experiments are performed on a biaxial tensile test platform to further verify the analytical model as shown in Fig. 6(a). The beam sample was manufactured with thermoplastic polylactic acid by three-dimensional (3D) printing technology, whose length (*l*), width (*b*), and thickness (*h*) are 150 mm, 20 mm, and 1 mm respectively. An extra length of 50 mm was spared for clamped portion, and it was clamped by two permanent magnets at one end. A rigid rod was assembled with one of its ends hinged to the beam and the other one going through a shaft sleeve, which can rotate freely at a fixed point. By pushing or pulling the rod, it will rotate with the deformed beam, which can stimulate the mechanism of the single-jack nozzle. All experiment results were photographed by a digital camera.

The deflections of the deformed beam were obtained by the tracing point method. Figures 6(b)–6(e) show the deflection results from the analytical model and experiments in circle symbols and solid lines, respectively. Figures 6(b) and 6(c) demonstrate the case of *θ*_{0} set as 0 and a displacement load of 50% and −50%, respectively. The cases with initial inclination angles of 30 deg and −30 deg are shown in Figs. 6(d) and 6(e), respectively. After removal of the loads, the beam recovered to its initial state, which indicated the beam in elastic range during experiments.

As seen from the above results, the theoretical and experimental results are basically identical, which demonstrates that the proposed method has a high accuracy to predict the deformation of the single-jack cantilever beam structure.

## 5 Conclusions

In the current work, we proposed a generalized vibrational method to solve the deformation problem of the single-jack flexible nozzle structure, which can be extended to deal with the large deformation beam problem with a movable hinge boundary. The FEA and experimental validations reveal the high accuracy and feasibility of the current method. It also provides a basis for solving the problem of multi-jack flexible nozzle structure and can serve as guidelines for the design of wind tunnels.

## Acknowledgment

This research was funded by the National Natural Science Foundation of China (NSFC) (Grant No. 12072150). This work is also supported by the Joint Fund of Advanced Aerospace Manufacturing Technology Research (Grant No. U1937601), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics, Grant No. MCMS-I-0221Y01), and the National Natural Science Foundation of China for Creative Research Groups (Grant No. 51921003).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*l*=length of the beam

*w*=deflection function

*P*=reaction force

*M*_{0}=bending moment at the left end of the beam

*S*_{0}=initial length of the jack

- EI =
bending stiffness

*α*=the length ratio of jack to beam

- Δ
*S*=variation length of the jack

*ɛ*_{app}=applied load

*θ*=rotation angle of the jack

*θ*_{0}=initial inclination angle of the jack

*λ*=Lagrange multiplier

*φ*=tangent angle at the right end of the beam